Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state

In this paper we prove that the focusing, d-dimensional mass critical nonlinear Schrödinger initial value problem is globally well-posed and scattering for u0∈L2(Rd), ‖u0‖L2(Rd)<‖Q‖L2(Rd), where Q is the ground state, and d≥1. We first establish an interaction Morawetz estimate that is positive definite when ‖u0‖L2(Rd)<‖Q‖L2(Rd), and has the appropriate scaling. Next, we will prove an interaction Morawetz estimate localized to low frequencies, similar to the estimates made in [13-15]. See also [12] for localization to high frequencies in the energy critical case.